Distributed Synchronization of Heterogeneous ... - CalTech Authors

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Distributed Synchronization of Heterogeneous Oscillators on Networks with Arbitrary arXiv:1405.6467v1 [math.OC] 26 May 2014

Topology Enrique Mallada, Randy A. Freeman, and Ao Tang

Abstract Many network applications rely on the synchronization of coupled oscillators. For example, such synchronization can provide networked devices with a common temporal reference necessary for coordinating actions or decoding transmitted messages. In this paper, we study the problem of using distributed control to achieve both phase and frequency synchronization of a network of coupled heterogeneous nonlinear oscillators. Not only do our controllers guarantee zero phase error in steady state under arbitrary frequency heterogeneity, but they also require little knowledge of the oscillator nonlinearities and network topology. Furthermore, we provide a global convergence analysis, in the absence of noise and propagation delay, for the resulting nonlinear system whose phase vector evolves on the n-torus. Index Terms Synchronization, coupled oscillators, control of networks, distributed control, nonlinear control.

I. Introduction Achieving temporal coordination among different networked devices is a fundamental requirement for the successful operation of many engineering systems. For example, it is necessary in communication systems for recovering transmitted messages [1], in sensor networks for coordinating wake up cycles [2] or achieving temporal measurement coherence [3], and in computer networks for preserving the causality of distributed events [4]. Almost ubiquitously, such coordination is accomplished by providing each node of the network with its own local oscillator and then compensating its phase and frequency (using information received from other devices on the network) to achieve a common temporal reference.

E. Mallada is with Caltech, CMS Dept. 1200 E California Blvd, Pasadena, CA 91125, USA, [email protected]. R. Freeman is with Northwestern U., EECS Dept., 2145 Sheridan Rd., Evanston, IL 60208-3118, USA, [email protected]. His work was supported in part by a grant from the Office of Naval Research. A. Tang is with Cornell U., School of ECE, Ithaca, NY 14853, USA, [email protected]. His work was supported in part by a grant from the Office of Naval Research.

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Legacy applications such as public switched telephone networks and cellular networks use a centralized hierarchical synchronization scheme with high-precision oscillators having relative frequency errors ranging from 0.01 to 4.6 parts per million (ppm) [5, 6]. For several reasons, however, these traditional synchronization architectures have become increasingly unsuitable for newer applications such as wireless sensor networks. For example, traditional methods can break down with the failure of only a few nodes. In addition, many newer applications use inexpensive oscillators having errors as high as 100 ppm [7]. Thus, a synchronization protocol designed for these newer applications should satisfy two essential requirements: it should be distributed and independent of the network topology (each node should use only its neighbors’ oscillator information to adjust it own oscillator), and it should be robust to wide variations and uncertainty in the specifications of the oscillators used throughout the network. A variety of synchronization algorithms have been proposed along these lines, jointly inspired by collective synchronization in physics and biology [8–11] and cooperative control in engineering networks [12, 13]. One possible solution is to use monotonically increasing time sources (e.g. clocks) and update their times based on offset information [14–22] to achieve a common absolute time reference (clock synchronization). This is suitable for applications in computer networks where a reference to an absolute time is needed (e.g. distributed databases). Another solution is to use periodic time sources (e.g. oscillators) interconnected with phase comparators [23–25] or pulse-coupling [26–28], where the objective is to achieve common relative time reference (phase synchronization) that allows temporal coordination within the network (e.g. waking up simultaneously).1 While the theoretical study of clock synchronization is fairly mature, with solutions that can provide zero offset error synchronization on networks with arbitrary heterogeneous frequencies [4] and asynchronous updates [22], little is known about the phase synchronization counterpart. For example, most phase synchronization solutions present nonzero steady state phase differences in the presence of frequency heterogeneity [23, 24, 26–28] with convergence guarantees limited to idealized scenarios such as homogeneous frequencies [29]. The only exception is [25] which can guarantee phase synchronization for complete graph topologies. Thus whether or not such systems can synchronize for arbitrary networks and arbitrary frequency heterogeneity has remained as an open question [24]. In this paper, we provide a positive answer to this question under very general conditions. We propose two distributed controllers that can achieve phase synchronization for a network of arbitrarily interconnected oscillators, under mild assumptions on the oscillator and phase comparator characteristics. For example, we allow the instantaneous frequency of each oscillator to be a highly uncertain nonlinear function of the local control input, a model consistent with most analog oscillators (such as voltage-controlled oscillators or 1Although one can use clock synchronization to achieve phase synchronization by simply mapping the linear times onto the circle using a modulo operator, this approach can lead to undesirable transients if the phases keep wrapping around the circle as the linear times synchronize. For this reason we consider these as separate synchronization problems, each suitable for different application areas.

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CMOS oscillators). Also, unlike existing work, we allow the set of oscillator frequencies to be bounded, so that each oscillator may operate within a prescribed frequency range, even during the transient part of the response. Finally, we allow flexibility in the choice of the phase comparator responses, rather than assuming as in [25] that they are sinusoidal. We only require that the measured phase difference is noiseless and can be obtained without propagation delay. The main contribution of the paper is a novel nonlinear convergence analysis that leverages recent results on the stability of equilibria of homogeneous-frequency coupled oscillators [29]. In particular, our controllers are based on a Hamiltonian dynamic system defined on the graph in which each local minimum of the energy function represents a synchronized trajectory. Each controller employs a different mechanism to dissipate energy and thereby converge to a synchronized solution. Furthermore, we show that any trajectories that are synchronized in frequency but not in phase must be unstable. II. Notation and terminology We let T = R/2πZ denote the unit circle, regarded as the Lie group of angle addition. We equip T with the usual Riemannian metric which defines the distance between two points to be the length of the shorter of the two arcs connecting them (so that diam(T) = π). For p ∈ N, we let Tp denote the Cartesian product of p circles. For i ∈ {1, . . . , p}, we let ∂i denote the unit vector field pointing in the counterclockwise direction on the ith factor of Tp (which we write simply as ∂ when p = 1). Because these unit vector fields form an ordered basis for the tangent space of Tp at each point, we can represent tangent vectors for Tp as elements of Rp , that is, as coordinate vectors with respect to this basis. Moreover, all Jacobian matrices of mappings defined on Tp will be representations of the differential with respect to this basis. All graphs in this paper will be simple, undirected, connected graphs having n vertices (with 2 6 n < ∞) and m edges (with m > n − 1). We represent such a graph G as a pair G = (V, E) for a vertex set V and edge set E. We label and order the vertices and edges, writing V = {1, . . . , n} and E = {1, . . . , m}, where each edge k ∈ E is an unordered pair of distinct vertices k = {i, j} ⊂ V. For each vertex i ∈ V, we let Ni denote the following indexed set of neighbors of i:  Ni = (j, k) ∈ V × E : k = {i, j} .

(1)

Thus (j, k) ∈ Ni if and only if (i, k) ∈ Nj , that is, if and only if edge k connects vertices i and j. III. Problem statement and results We consider a network of controlled oscillators in which each oscillator shares the current value of its phase with its immediate neighbors. The purpose of the controller design is to guarantee both frequency and phase synchronization of the interconnected system. We adopt the classical phase-locked loop (PLL) structure for each controlled oscillator [23, 30]. This structure consists of three components connected in feedback, as illustrated in Fig. 1: a base oscillator, a phase comparator, and a loop filter. The base oscillator is a physical device (such as a voltage-controlled oscillator) whose frequency is determined dynamically by

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control signal loop filter

base oscillator

phase error

phase phase comparator ···     

   

neighbors’ phases

controlled oscillator

Fig. 1: PLL components of a controlled oscillator.

means of a control signal; the phase of each controlled oscillator system is simply the phase of its base oscillator. The phase comparator produces a phase error signal by comparing its own phase with the phases of its neighbors. Finally, the loop filter produces the control signal from the phase error. We represent the network of oscillators by a graph G = (V, E) in which each vertex is a controlled oscillator and each edge indicates an exchange of phases between neighboring vertices. We represent the phase of oscillator i at time t ∈ R by ϕi (t) ∈ T, and we write the phase vector signal for the entire network as the column vector  T ϕ = ϕ1 . . . ϕn ∈ Tn .

(2)

We define a synchronization measure δϕ to be the diameter of the finite subset of T consisting of the n phases ϕi :  δϕ = diam ϕ1 , . . . , ϕn ∈ [0, π] . Note that if δϕ
0 and each i ∈ V; internal boundedness: all loop filter states (if any) are bounded in forward time. The constraint interval I used in this definition characterizes the desired range of frequencies for the oscillators. Choosing an appropriate interval I in the design is thus useful for preventing the oscillators

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from moving too fast, too slow, or reversing direction. A. Base oscillator and phase comparator In common models of controlled analog oscillators, the instantaneous frequency is some uncertain monotonic nonlinear function of the control input. For example, the graph of this function is the “tuning curve” seen on data sheets of many voltage-controlled oscillators, where it is understood that this curve is typical rather than exact. Motivated by such models, we assign an uncertain frequency function χi to the base oscillator in each vertex i, where χi : U → R is a strictly increasing C 1 function on a known open interval domain U ⊂ R. The angular velocity of the phase of the oscillator at time t (that is, its instantaneous frequency) is then given by χi (ui (t)), where ui (·) is a U-valued control signal. Hence our model for the base oscillator is the differential equation ϕ˙ i = χi (ui ) .

(4)

In this oscillator model, the oscillator stops whenever χi (ui ) = 0 and reverses direction whenever χi (ui ) changes sign. However, an oscillator need not admit such behavior, as the image χi (U) need not contain zero. As we see from the model (4), the base oscillators are nonlinear and heterogeneous. In addition, we do not need precise knowledge of the frequency functions χi to complete our design and guarantee almost-global synchronization. In fact, as we will see when we list our assumptions in Section III-E below, all we need to know about the functions χi in the unconstrained case I = R is that they are C 1 with positive derivatives on U, and that the intersection of their images χi (U) is nonempty. The phase comparator in vertex i calculates a linear combination of functions of phase differences to produce a phase error ei : ei (ϕ) =

X

ak f (ϕj − ϕi ) ,

(5)

(j,k)∈Ni

where the constants ak are positive edge weights and f : T → R is the phase coupling function. For example, the coupling function used in many classical PLL designs is the sine function f = sin. B. Example: the Kuramoto model Suppose that the frequency function χi for each base oscillator i has the simple affine form χi (ui ) = ωi +ui , where the constant ωi ∈ R represents the nominal oscillator frequency. Suppose further that the phase coupling function is the sine function f = sin and that all loop filters are constant unity gains so that ui ≡ ei . Then the controlled phase equation becomes X ϕ˙ i = ωi + ak sin(ϕj − ϕi ) .

(6)

(j,k)∈Ni

This model of a network of coupled oscillators has been studied extensively, and we refer the reader to the survey paper [31]. In particular, if the graph G is complete, and if all edge weights ak are the same, then this is the famous Kuramoto model of coupled oscillators [32].

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The oscillator network characterized by the dynamics in (6) fails to meet the design goal of Definition 1. Indeed, the existence of a synchronized trajectory for the model (6) in which δϕ ≡ 0 implies that the nominal frequencies ωi are all identical. If these frequencies were known precisely, then we could simply cancel them out via control by setting ui = ei + ω ? − ωi to obtain a model of the form (6) in which ωi = ω ? for all i ∈ V. However, even in this case of identical frequencies, almost-global synchronization has been proved only for special classes of connected graphs, such as complete graphs or trees [31]. Instead, we are interested in oscillators having unknown heterogeneous frequencies on arbitrary connected graphs. In this case one can choose sufficiently large edge weights ak to guarantee “practical” phase synchronization in which δϕ becomes small, provided it does not start off too large [31]. However, choosing large edge weights makes it less likely that the frequencies ϕ˙ i will be constrained to a desired interval I during the transient. In any case, we see that we must depart from this standard model (6) to meet the design goal of Definition 1, and we do so by choosing a non-sinusoidal coupling function f in (5) and a nonlinear dynamic loop filter. C. Toward synchronization: a Hamiltonian system The loop filter in vertex i produces the control signal ui from the phase error ei . It is well known that introducing integral action into the loop filter can compensate offset mismatches for networks of heterogeneous oscillators [19, 21–23, 25, 30]. Thus as a first attempt at achieving almost-global synchronization of the oscillator network, we simply make the loop filter a scaled integrator: γ˙ i = ci ei (ϕ)

(7)

ui = ζ(γi ) ,

(8)

where γi (·) is the real-valued internal filter state, ci is a positive parameter, and ζ : R → U is a scaling function which squeezes the value of γi into the domain U of the frequency function χi . We do not assume that the oscillators have access to the global time variable t. As a result, the differential dt used in the construction of dγi /dt in (7) is unknown, and we account for this by assuming that the positive constant ci is unknown. The system resulting from (4) and (7)–(8) is ϕ˙ i = χi (ζ(γi ))

(9)

γ˙ i = ci ei (ϕ) ,

(10)

where ei is the phase error in (5). The first thing we notice about this system (9)–(10) is that it admits a synchronized solution at any frequency ω ? ∈ R which belongs to the image of each function χi ◦ ζ. Indeed, choose any initial states such that δϕ(0) = 0 and χi (ζ(γi (0))) = ω ? for each i ∈ V; then ϕ˙ i ≡ ω ? and γ˙ i ≡ 0 for every i ∈ V, which implies δϕ ≡ 0. However, solutions starting from other initial conditions will not converge to such synchronized trajectories, which means this system still fails to meet the design goal of Definition 1.

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The second thing we notice about the system (9)–(10) is that it is a Hamiltonian system with position variables ϕi and momentum variables γi /ci . Indeed, assuming the phase coupling function f is odd, the n-vector of phase errors ei is the negative gradient of a potential function of the phase vector ϕ (as we will show in Section IV-A). Furthermore, each function (χi ◦ ζ)/ci , being a scalar function of a scalar variable, is trivially the gradient of a potential function of its argument γi . The sum of these potential functions is a Hamiltonian associated with the dynamics (9)–(10). Moreover, as we will see in Section IV-A, if we let the position variables be the phases ϕi measured relative to an appropriate rotating frame, then the system is still Hamiltonian, but now the Hamiltonian is proper and nonnegative and is thus a Lyapunov function candidate. If we can guarantee that synchronized trajectories represent the only local minima of this Lyapunov function, then we can perturb these Hamiltonian dynamics with dissipation terms in the controller to achieve almost-global synchronization. This is the control design strategy we will pursue in this paper. D. The PI loop filter: a perturbed Hamiltonian system We propose two different perturbed Hamiltonian systems in this paper, the first now and the second later on in section III-F. For our first perturbation, we add a proportional term to the integral control (8), resulting in a proportional-integral (PI) loop filter of the form γ˙ i = ci ei (ϕ)

(11)

ui = ζ(ei (ϕ) + γi ) .

(12)

Such a loop filter (without our nonlinear scaling function ζ) can be found in classical PLL designs [30] as well as in various coupled oscillator systems [23, 25]. More generally, the error term ei in (12) can be replaced with a scaled error term κi ei , where the “proportional gain” κi is a positive constant (or even a positive function of γi ), with virtually no change in the convergence proof. We have left this gain κi out of the analysis to simplify notation, but the flexibility it adds will be important for tuning the performance of the system. We might also introduce a corresponding “integral gain,” but for the analysis this can be absorbed into the constant ci . To summarize, each controlled oscillator in the network has second-order dynamics of the form ϕ˙ i = χi (ζ(ei (ϕ) + γi ))

(13)

γ˙ i = ci ei (ϕ)

(14)

with ei (ϕ) =

X

ak f (ϕj − ϕi ) .

(15)

(j,k)∈Ni

The state space of each such oscillator is the cylinder T× R. We next present conditions under which this oscillator system exhibits almost-global synchronization within a given interval I.

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E. Assumptions and main result We assume that each vertex knows its neighbors in the graph G (as they need to exchange phase information), but otherwise the graph is unknown. However, we do assume the following: (A1) the graph G is connected, and there exists a known upper bound on the number n of vertices in G. The frequency functions χi of the base oscillators are uncertain; we merely assume that they satisfy the following: (A2) the frequency functions χi : U → R are all C 1 with positive derivatives χ0i : U → (0, ∞), and are such that \

χi (U) 6= ∅ .

(16)

i∈V

It is clear from the oscillator model (4) that (16) is necessary for the existence of phase trajectories having synchronized frequencies. This condition (16) implies the existence of some common interval of possible base oscillator frequencies, and thus places an inherent limit on the extent to which these oscillators can differ from each other. Indeed, if one oscillator can only produce frequencies in the kilohertz range and another only in the megahertz range, then there is no possibility of synchronization. We next assume that our loop filter scaling function ζ satisfies: (A3) the scaling function ζ : R → U is C 1 with positive derivative ζ 0 : R → (0, ∞), and is such that \

χi (ζ(R)) 6= ∅

(17)

χi (ζ(R)) ⊂ I ,

(18)

i∈V

and [ i∈V

where I is the constraint interval from Definition 1. It is clear from (4) that (18) constrains the frequencies ϕ˙ i to the interval I as required by the design goal in Definition 1. If I is large enough to contain the union of the images χi (U), then we can always satisfy assumption (A3) by choosing the scaling function ζ to be a diffeomorphism onto U (so that (16) and (17) are the same). However, if I is not that large, then assumption (A3) states that we have found a solution to the problem of designing ζ to satisfy both (17) and (18) based on some a priori knowledge about the set of possible frequency functions χi . Such a design problem could very well have no solution if I is too small. We let f 0 : T → R denote the derivative of the phase coupling function f in the direction of the unit vector field ∂. We make two assumptions on this function f : (A4) the phase coupling function f : T → R is C 1 and odd, that is, f (−θ) = −f (θ) for all θ ∈ T; π (A5) there is a constant b ∈ (0, n−1 ] such that f 0(θ) > 0 whenever cos(θ) > cos(b) and f 0(θ) < 0 whenever

cos(θ) < cos(b), for any θ ∈ T.

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tanlock degree p = 3 degree p = 6 degree p = 20

0.4 0.2 0 −0.2 −0.4 −π

−b

0

b

π

Fig. 2: Examples of phase-coupling functions f when b = 0.5.

Assumptions (A4)–(A5) also appeared in [33, 34] and play an important role in the convergence analysis: (A4) allows the interpretation of (4) and (7)-(8) as a Hamiltonian system, while (A5) is needed to guarantee convergence to the desired solution. Note that to choose such a parameter b, we must use our assumed knowledge in (A1) of a known upper bound on n. Examples of functions f which satisfy assumptions (A4)–(A5) are shown in Fig. 2 for b = 0.5. The first example is given by the C ω formula   f (θ) = 1 − cos(b)

sin(θ) . 1 − cos(b) cos(θ)

(19)

This function is related to the characteristic of certain “tanlock” phase comparators [30], and it generates the sinusoidal coupling f = sin when b =

π 2.

The other examples are C 1 and piecewise polynomial of

various degrees p > 1, each having a derivative given by f 0(θ) = 1 − (|θ|/b)p−1 on a certain arc containing [−b, b] and a constant derivative on its complement. All of these examples are normalized to have unit derivative at zero, which means they should result in similar performance for small deviations around a stable synchronized trajectory. When b is small (which we require when n is large), the magnitude of the derivative of the tanlock function is small on the arc [b, π] when compared to the magnitude of the derivatives of the piecewise-polynomial functions. As a result, the piecewise-polynomial functions might provide faster convergence to a synchronized state when some initial phase differences are greater than b (due to their larger gains for large phase differences). The final assumption is on the choice of the edge weights ak : (A6) each edge weight ak is chosen at random from a continuous probability distribution on the interval (0, ∞). This assumption allows us to state that, with probability one, we avoid an unknown zero-measure set of bad edge weight vectors a = [a1 . . . am ]T ∈ Rm for which our stability analysis does not guarantee convergence. Theorem 2. Assume (A1)–(A6). Then with probability one in the selection of edge weights in (A6), the network of oscillators with vertex dynamics (13)–(15) achieves almost-global synchronization within I.

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M leaf

leaf

Fig. 3: The invariant set M (shown in gray) is a collection of isolated cylinders. The intersection of each invariant leaf of the foliation (shown in blue) with M is a collection of isolated circles (shown in red).

Note that when n > 4, assumption (A5) rules out the sinusoidal phase-coupling function f = sin. In fact, the requirement b 6 b>

2π n ,

π n−1

is within a factor of two of being necessary. Indeed, one can show that if

then for some connected graphs having n vertices there exists a nonempty open set of bad edge

weight vectors such that trajectories of (13)–(15) starting from a set of initial states having positive measure achieve asymptotic frequency synchronization but not asymptotic phase synchronization. We will prove Theorem 2 in Section IV. In this proof, we first construct a Lyapunov function for the system, and then apply the Krasovskii-LaSalle invariance theorem to show that all trajectories converge to a certain invariant subset M of the state space Tn × Rn . We then determine the structure of this set M, showing first that it is the disjoint union of isolated embeddings of the cylinder T× R. We next show that the state space admits a foliation into invariant (2n−1)-dimensional submanifolds, and that the intersection of each leaf of this foliation with the set M is the disjoint union of isolated embeddings of the circle T (as conceptualized in Fig. 3). Each such circle represents a periodic trajectory with a constant synchronized frequency ω ? ∈ I (so that ϕ˙i ≡ ωi? ). One of the cylinders of the set M is good, in that all of the periodic trajectories on this cylinder are also synchronized in phase (so that δϕ ≡ 0). The remaining cylinders of M (if any) are bad, in that all of the periodic trajectories on these cylinders are out of phase. Finally, using a local linearization analysis, we show that the phase-synchronized trajectories are exponentially stable (relative to each leaf of the foliation), whereas the out-of-phase trajectories are exponentially unstable.

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Because we show that these periodic trajectories are isolated on each leaf (a key step in the proof), we conclude that almost all trajectories achieve asymptotic synchronization in both frequency and phase. F. The dual controller: a different perturbed Hamiltonian system The approach we take in the controller design in (13)–(15) is to perturb the Hamiltonian system (9)–(10) by adding a dissipation term to the dynamics of each position variable. An alternative, dual approach is to add a dissipation term to the dynamics of each momentum variable. We will do this by including a second comparator in addition to the phase comparator, that is, by introducing a second set of error signals zi in addition to the phase errors ei in (5). The resulting controller is more complex, but further analysis might reveal that it has performance advantages over the controller (13)–(15). For this dual controller, each pair of neighboring vertices agrees on an orientation of the edge connecting them. Thus each vertex i can partition its neighbor set as Ni = Ni+ ∪ Ni− , where (j, k) ∈ Ni+ when i is the head of the edge k = {i, j} and (j, k) ∈ Ni− when i is the tail of the edge k. Furthermore, we make an additional assumption on the frequency functions χi of the base oscillators: (A7) the interval U and the intervals χi (U) for each i ∈ V contain only positive real numbers. In other words, we assume that the graph of each χi (·) lies completely in the first quadrant, which is true for many analog oscillators. This assumption guarantees that the functions ζ and χi take on only positive values. Consequently, for each ordered pair of vertices ij we can define the ratio ρij (γi , γj ) =

ζ(γj )χi (ζ(γi )) > 0. ζ(γi )χj (ζ(γj ))

(20)

Using (9), that is, using the base oscillator model (4) together with the integrator loop filter (7)–(8), we can calculate (20) as ρij (γi , γj ) =

ζ(γj ) h dϕj i−1 . ζ(γi ) dϕi

(21)

We then define a new error signal zi for vertex i as zi (γ) =

X

  dk ζ(γj ) − ρij (γi , γj )ζ(γi )

(j,k)∈Ni+

+

X

  dk ρ−1 ij (γi , γj )ζ(γj ) − ζ(γi ) ,

(22)

(j,k)∈Ni−

where γ = [γ1 . . . γn ]T ∈ Rn is the vector of loop filter states and the constants dk are an additional set of positive edge weights. For vertex i to compute this error variable zi , it must receive the signal ζ(γj ) from each of its neighbors. Also, we see from (21) that it must calculate the derivative of each neighbor’s phase ϕj with respect to its own phase ϕi , a calculation which does not require knowledge of the global time variable t.

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Using these new error signals zi , we define the dual perturbed Hamiltonian system as ϕ˙ i = χi (ζ(γi ))

(23)

γ˙ i = ci ei (ϕ) + ci zi (γ) .

(24)

Thus we have retained the simple integrator loop filter in (7)–(8), but have changed its input from ei to the sum ei + zi . This amounts to modifying Fig. 1 by putting a “γ-comparator” in parallel with the phase comparator. Although the at first sight the physical interpretation of zi is obscured by the implementation details, substituting (20) and (23) into (22) gives zi (γ) =

X

  ηk (γ) χj (ζ(γj )) − χi (ζ(γi ))

(j,k)∈Ni

=

X

  ηk (γ) ϕ˙ j − ϕ˙ i

(25)

(j,k)∈Ni

with ηk (γ) = dk

ζ(γj ) , where j is the tail of edge k, χj (ζ(γj ))

(26)

for each k ∈ E. Therefore (22) can be interpreted as an indirect estimate of the frequency mismatch (25) between oscillator i and its neighbors. As we will see in the proof of the following theorem in Section V below, this frequency error terms zi are the right perturbation that need to be added to the dynamics of the momentum variables to guarantee the dissipation of the Hamiltonian energy. Theorem 3. Assume (A1)–(A7). Then with probability one in the selection of edge weights in (A6), the network of oscillators with vertex dynamics (22)–(24) achieves almost-global synchronization within I. The proof of this theorem is similar to that of Theorem 2, and we will highlight the main differences in Section V. IV. Proof of Theorem 2 For any p ∈ N, we let 0p and 1p denote the column vectors of p zeros and p ones, respectively, and we let Ip denote the p × p identity matrix. Any ` × p matrix with integer elements will represent either an R-linear map from Rp to R` or a Z-linear map from Tp to T` , depending on context. In particular, if M is an `×p matrix with integer elements and t 7→ x(t) is a curve in Tp , then t 7→ y(t) = M x(t) is a curve in T` (with M representing a Z-linear map), and furthermore y(t) ˙ = M x(t) ˙ for all t (with M now representing an R-linear map). We first introduce some notation for writing the overall coupled dynamics (13)–(15) in a compact form.

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We define state vectors  T ϕ = ϕ1 . . . ϕn  T γ = γ1 . . . γn

∈ Tn

(27)

∈ Rn

(28)

A = diag{a1 , . . . , am }

∈ Rm×m

(29)

C = diag{c1 , . . . , cn }

∈ Rn×n .

(30)

along with the following diagonal matrices:

For convenience we define σi = χi ◦ ζ for each i, and we note from assumptions (A2)–(A3) that each σi is C 1 with a positive derivative. We next define mappings F : Tm → Rm and Σ : Rn → Rn by h iT F (θ) = f (θ1 ) . . . f (θm ) h iT Σ(y) = σ1 (y1 ) . . . σn (yn )

∈ Rm

(31)

∈ Rn ,

(32)

where θ = [θ1 . . . θm ]T ∈ Tm and y = [y1 . . . yn ]T ∈ Rn . Note that Σ is a C 1 diffeomorphism onto its image Σ(Rn ). Finally, we let B ∈ {−1, 0, 1}n×m be an oriented incidence matrix for the graph G. Using the fact from (A4) that f is odd, we can write the dynamics (13)–(15) as ϕ˙ = Σ(−BAF (B Tϕ) + γ)

(33)

γ˙ = −CBAF (B Tϕ) .

(34)

For convenience we define e : Tn → Rn by  T e(ϕ) = e1 (ϕ) . . . en (ϕ) = −BAF (B Tϕ) ,

(35)

which enables us to write (33)–(34) as ϕ˙ = Σ(e(ϕ) + γ)

(36)

γ˙ = Ce(ϕ) .

(37)

Note that 1Tn e ≡ 0 because 1Tn B = 0 (a property of any oriented incidence matrix). This property also implies e(ϕ) = e(ϕ + 1n θ) for any ϕ ∈ Tn and any θ ∈ T. A. Global Lyapunov function In this section we construct a Lyapunov function for the system (36)–(37), which is the Hamiltonian we described in Section III-C for the system (9)–(10). Because f is odd from (A4), the integral of the 1-form f · h∂, ·i around any smooth closed curve in T is zero. Thus this 1-form is the differential of a smooth function Ψ : T → R, which is unique up to an additive constant (which we choose so that the minimum value of Ψ on T is zero). Therefore

d dt (Ψ ◦ x)

≡ f (x)x˙ for

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any curve x : R → T. We then define V : Tm → [0, ∞) as the sum X ak Ψ(θk ) , V (θ) =

(38)

k∈E

where θ = [θ1 . . . θm ]T ∈ Tm . It follows that d V (B Tϕ) = F T (B Tϕ)AB T ϕ˙ = −eT (ϕ)ϕ˙ . (39) dt T From (17) there exists a frequency $ ∈ i σi (R). Therefore the function W : Rn → [0, ∞) defined as X 1 Z γi   σi (s) − $ ds (40) W (γ) = ci σi−1 ($) i∈V

is proper (because σi is increasing) and has a derivative given by   d W (γ) = ΣT (γ) − $1Tn C −1 γ˙ . dt

(41)

We now obtain a Lyapunov function by adding (38) and (40): we define U : Tn × Rn → [0, ∞) by U (ϕ, γ) = V (B Tϕ) + W (γ) ,

(42)

which is a proper function on Tn×Rn . Because 1Tn e ≡ 0, the derivative of (42) along trajectories of (36)–(37) is U˙ = −eT (ϕ)Σ(e(ϕ) + γ) + ΣT (γ)e(ϕ) Z 1 T = −e (ϕ) Σ0(e(ϕ)s + γ) ds · e(ϕ) ,

(43)

0

where Σ0(·) denotes the diagonal Jacobian matrix  Σ0(γ) = diag σ10 (γ1 ), . . . , σn0 (γn ) .

(44)

Because Σ0(·) is positive definite, we have U˙ 6 0 and U˙ = 0 if and only if e(ϕ) = 0. It follows from the Krasovskii-LaSalle invariance theorem that all trajectories of the system (36)–(37) converge to the largest invariant set M contained within the set Φ × Rn , where Φ ⊂ Tn denotes the set  Φ = e−1 ({0}) = ϕ ∈ Tn : BAF (B Tϕ) = 0 .

(45)

Note that because B T 1n = 0, this set Φ has the symmetry property Φ = Φ + 1nT. Also, it follows from (A4) that F (0) = 0, which means Φ contains all points of the form 1n θ for θ ∈ T. The next step in the proof is to investigate the structure of this set M. B. Structure of the set M In this section we show that the largest invariant set M contained within the set Φ × Rn is M = Φ × Γ, where Φ in (45) is the zero set of e and Γ ⊂ Rn is the set  Γ = γ ∈ Rn : B T Σ(γ) = 0 .

(46)

We begin by exploring the structure of these sets Φ and Γ. First, we show that Φ is the disjoint union of isolated embeddings of the circle T. Next, we show that Γ = α(R), where α : R → Rn is a C 1 curve in Rn .

15

Moreover, if we let q ∈ Rn denote the unit vector in the direction of C −1 1n , then this curve α is such that q T α(·) is the identity map on R. Finally, we show that M = Φ × Γ, that is, that M is the disjoint union of isolated embeddings of the cylinder, as illustrated in Fig. 3. We partition Tn using the following matrices:   0Tn−1  R= In−1   −1Tn−1  S= In−1

∈ {0, 1}n×(n−1)

(47)

∈ {−1, 0, 1}n×(n−1) .

(48)

Clearly the state ϕ satisfies the identity ϕ = 1n ϕ1 + RS Tϕ ,

(49)

which defines the direct sum Tn = 1nT⊕RTn−1 . Here the first summand represents the first component angle and the second one represents the remaining angles measured relative to the first. Note that S TR = In−1 , that S T 1n = 0, and that B T = B TRS T . Also, because G is connected from (A1), we have rank(B) = n − 1 and thus the columns of B TR are independent. Because ϕ ∈ Φ if and only if RS Tϕ ∈ Φ, it follows from (49) that Φ = RS T Φ + 1nT. It also follows that for any µ ∈ Tn−1 , we have µ ∈ S T Φ if and only if Rµ ∈ Φ, that is, if and only if e(Rµ) = 0. We next show that the points in the set S T Φ are isolated, which implies that Φ is the disjoint union of isolated embeddings of the circle T. Using the above partition of Tn , we define two symmetric matrix functions L : Tn−1 → Rn×n and L[ : Tn−1 → R(n−1)×(n−1) by L(µ) = BAF 0(B TRµ)B T

and

L[ (µ) = RT L(µ)R

(50)

for µ ∈ Tn−1 , where F 0(·) denotes the diagonal Jacobian matrix  F 0(θ) = diag f 0(θ1 ), . . . , f 0(θm )

(51)

with θ = [θ1 . . . θm ]T ∈ Tm . Here L(µ) represents a weighted Laplacian matrix for the graph G in which the weights can have positive, negative, or zero values. Also, because B = SRT B we have L(µ)R = SL[ (µ)

and

L(µ) = SL[ (µ)S T

(52)

for all µ ∈ Tn−1 . Note that L(µ) is congruent to the block diagonal matrix diag{0, L[ (µ)}. The proof of the following theorem is in Appendix A: Theorem 4. There is a set Z ⊂ Rm having zero Lebesgue measure such that if a = [a1 . . . am ]T 6∈ Z, then the matrix L[ (µ) in (50) is invertible for all µ ∈ S T Φ. Corollary 5. If a 6∈ Z then the points in S T Φ are isolated.

16

Proof: Define the mapping P : Tn−1 → Rn−1 by setting P (µ) = RTBAF (B TRµ) so that P −1 ({0}) = S T Φ. The Jacobian matrix for P is just L[ (µ), which by Theorem 4 is invertible for all µ ∈ S T Φ. The result follows from the inverse function theorem. We do not provide a method for computing the set Z of bad edge weight vectors; instead, we rely on the random edge weight selection in (A6) to avoid this zero-measure set. It is clear from (45) and (50) that this set Z depends only on the graph G (through B) and the phase coupling function f (through F ). In principle we could remove the dependence on the unknown graph by taking the countable union of all such sets Z over all connected graphs to be our zero-measure set of bad edge weights, but this larger set would still be difficult to compute. Thus from now on we will assume a 6∈ Z, which happens with probability one according to (A6) To explore the structure of the set Γ in (46), we partition Rn by defining q ∈ Rn and Q ∈ Rn×(n−1) as q=

C −1 1n kC −1 1n k

Q = CS S T C 2 S

and


− 12

.

(53)

Using the fact that 1Tn S = 0, it is straightforward to show that the n × n matrix [q Q] is orthogonal; thus the state γ satisfies the identity γ = qq T γ + QQT γ ,

(54)

which defines the direct sum Rn = qR ⊕ QRn−1 via orthogonal projections. We define the open interval J ⊂ R as J=

\

σi (R) ,

(55)

i∈V

which is nonempty from (17). Because s1n ∈ Σ(Rn ) for all s ∈ J, we can define the function β : J → R as β(s) =

X

qi σi−1 (s) = q T Σ−1(s1n ) ,

(56)

i∈V

where the constants qi are the components of the vector q. Note that each term in the sum in (56) is strictly increasing in s. Suppose {s` } is a sequence in J such that s` → sup J as ` → ∞. Then there exists i ∈ V such that s` → sup σi (R) as ` → ∞, which means σi−1 (s` ) → ∞ as ` → ∞. Thus β(s` ) → ∞ as ` → ∞, which means sup β(J) = ∞. Similar reasoning yields inf β(J) = −∞, and we conclude that β is invertible. Therefore we can define a C 1 curve α : R → Rn as follows: α(r) = Σ−1 (β −1 (r)1n ) ,

(57)

which satisfies q T α(r) = β(β −1 (r)) = r for all r ∈ R. Because rank(B) = n − 1, it follows from (46) that γ ∈ Γ if and only if Σ(γ) = x1n for some x ∈ J, that is, if and only if γ = Σ−1 (x1n ) = Σ−1 (β −1 (β(x))1n ) = α(β(x)) for some x ∈ J. Therefore Γ = α(β(J)) = α(R).

(58)

17

Our remaining task in this section is to show that M = Φ × Γ. First we calculate the derivative of e(ϕ) in (35) using (36): e˙ = −BAF 0(B Tϕ)B T Σ(e(ϕ) + γ) = −L(S Tϕ) Σ(e(ϕ) + γ) .

(59)

We see from (36)–(37) that ϕ˙ = Σ(γ) and γ˙ = 0 on the set Φ × Rn , which means the second derivative ϕ¨ is zero: 0 = ϕ¨ = −Σ0(γ)L(S Tϕ) Σ(γ) = −Σ0(γ)SL[ (S Tϕ)S T Σ(γ)

(60)

on Φ × Rn . Now Σ0(·) in (44) is positive definite, L[ (S Tϕ) is invertible from Theorem 4, and S in (48) has independent columns; therefore (60) implies S T Σ(γ) = 0. Because B T = B TRS T , this in turn implies B T Σ(γ) = 0, and we conclude that M ⊂ Φ×Γ. We observe that Φ×Γ is itself invariant under the dynamics (36)–(37), and it follows that M = Φ × Γ. To summarize the results of this section, we have found that (ϕ, γ) ∈ M if and only if e(ϕ) = 0 and γ = α(r) for some r ∈ R, in which case r = q T γ. Furthermore, the zero set of e in (45) is RS T Φ + 1nT, where the points in S T Φ are isolated. C. Global analysis: frequency synchronization To study the synchronization properties of our system (36)–(37), we first observe that q T γ˙ ≡ 0, which means the state space admits a foliation whose leaves are the invariant manifolds Tn × Ξr , where  Ξr = qr + QRn−1 = γ ∈ Rn : q T γ = r

(61)

for a constant parameter r ∈ R. Note that because Γ = α(R) and q T α(·) is the identity map, the intersection Γ ∩ Ξr is the singleton {α(r)}. Thus the intersection of each invariant leaf Tn × Ξr with M = Φ × Γ is just Φ × {α(r)}, which we have shown to be the disjoint union of isolated embeddings of the circle T (depicted as red circles in Fig. 3). We now fix r ∈ R and examine the dynamics on Tn × Ξr , noting that γ ≡ qr + QQT γ on this invariant manifold. Now that r is fixed, we will write αr = α(r). Because αr = qr + QQT αr , we have γ − αr ≡ QQT (γ − αr )

(62)

on Tn × Ξr . We next define the projected state variables w1 = S Tϕ ∈ Tn−1

and

w2 = QT (γ − αr ) ∈ Rn−1 .

(63)

Taking time derivatives of these variables, and using (62) and the fact that e(ϕ) = e(Rw1 ), we obtain w˙ 1 = S T Σ(e(Rw1 ) + Qw2 + αr )

(64)

w˙ 2 = QTCe(Rw1 ) .

(65)

18

This is an autonomous system in the projected states (w1 , w2 ), and its equilibria are precisely all points of the form (µ? , 0) for vectors µ? ∈ S T Φ. Each equilibrium represents a frequency-synchronized solution of (36)–(37) with ϕ˙ ≡ β −1 (r)1n and γ ≡ αr . The equilibrium with µ? = 0 represents a phase-synchronized trajectory with δϕ ≡ 0, and all other equilibria represent out-of-phase trajectories. Furthermore, each trajectory of the system (64)–(87) converges to an equilibrium, which means each trajectory of the system (36)–(37) achieves asymptotic frequency synchronization. Indeed, we have shown that all trajectories of the system (36)–(37) converge to the set M = Φ × Γ in forward time. Therefore γ converges to the singleton set Γ ∩ Ξr = {αr }, which means w2 converges to zero. In addition, ϕ converges to the set Φ, which means w1 converges to the set S T Φ. Corollary 5 states that points in S T Φ are isolated, and we conclude that w1 converges to one of these points. In the next section, we will perform a local linearization analysis at each equilibrium of the system (64)–(87) to determine its stability. D. Local analysis: phase synchronization We compute the linear approximation of the dynamics (64)–(87) at an equilibrium (µ? , 0) as follows: w˙ 1 ≈ −S T Σ0(αr )L(µ? )R(w1 − µ? ) + S T Σ0(αr )Qw2

(66)

w˙ 2 ≈ −QTCL(µ? )R(w1 − µ? ) ,

(67)

w˙ 1 ≈ −S T Σ0(αr )SL[ (µ? )(w1 − µ? ) + S T Σ0(αr )Qw2

(68)

w˙ 2 ≈ −QTCSL[ (µ? )(w1 − µ? ) .

(69)

or using (52),

If we define the (n − 1) × (n − 1) matrices X = S T Σ0(αr )S > 0 Y = QTCS = S T C 2 S

(70)
12

>0

Z = S T Σ0(αr )Q ,

(71) (72)

then we can write this approximation more compactly as w˙ 1 ≈ −XL[ (µ? )(w1 − µ? ) + Zw2

(73)

w˙ 2 ≈ −Y L[ (µ? )(w1 − µ? ) .

(74)

Note from (53) and (71) that Q = CSY −1 so that S = C −1 QY , which means Z in (72) satisfies Y −1 Z = QTC −1 Σ0(αr )Q > 0 .

(75)

Thus the linearization (73)–(74) satisfies the assumptions in the following theorem, whose proof is in Appendix B:

19

Theorem 6. Let Λ ∈ R2p×2p be the block matrix   −XL Z , Λ= −Y L 0

(76)

where L, X, Y, Z ∈ Rp×p satisfy: 1) L is symmetric, 2) X + X T > 0, 3) Y is symmetric and invertible, and 4) Y −1Z is symmetric with Y −1Z > 0. If L has a negative eigenvalue, then Λ has an eigenvalue with a positive real part. If instead Z is invertible and L > 0, then Λ is Hurwitz. We can complete the local linearization analysis by applying Theorem 6 together with the following theorem: Theorem 7. The Laplacian L(µ) has n − 1 positive eigenvalues for µ = 0, and it has at least one negative eigenvalue for any nonzero µ ∈ S T Φ. The proof of Theorem 7, which can be found in [29], relies on the properties of the phase coupling function f defined in Assumption (A5). The basic intuition is that for µ? 6= 0, choosing b small enough guarantees a negative eigenvalue in L(µ? ) independently of the choice of µ? ∈ S T Φ. Let us now consider an equilibrium (µ? , 0) of the nonlinear system (64)–(87). Recall that L(µ? ) is congruent to diag{0, L[ (µ? )}; thus Theorem 7 together with Sylvester’s law of inertia imply that L[ (µ? ) > 0 when µ? = 0 and that L[ (µ? ) has a negative eigenvalue for all nonzero µ? ∈ S T Φ. Therefore if µ? = 0, which represents an in-phase steady-state solution, then it follows from Theorem 6 that this equilibrium is exponentially stable. Likewise, if µ? 6= 0, which represents an out-of-phase steady-state solution, then this equilibrium is exponentially unstable. Because all out-of-phase equilibria are both isolated and exponentially unstable, we conclude (say from [35, Proposition 1], for example) that the set Sr ⊂ Tn × Ξr of initial states from which trajectories converge to out-of-phase steady-state solutions has zero measure in Tn×Ξr (regarded S here as a (2n − 1)-dimensional manifold). It then follows from Tonelli’s theorem that the set S = r∈R Sr has zero measure in Tn × Rn . In other words, the system achieves asymptotic phase synchronization from almost every initial state. V. Proof of Theorem 3 We define the diagonal matrix function  H(γ) = diag η1 (γ), . . . , ηm (γ)

∈ Rm×m ,

(77)

20

so that we can write the dynamics (23)–(24) as ϕ˙ = Σ(γ)

(78)

γ˙ = Ce(ϕ) − CBH(γ)B T Σ(γ) .

(79)

Because 1Tn e ≡ 0 = 1Tn B, the derivative of (42) along trajectories of (78)–(79) is U˙ = −ΣT (γ)BH(γ)B T Σ(γ) .

(80)

Now H(·) is positive definite; hence U˙ 6 0 and U˙ = 0 if and only if B T Σ(γ) = 0. It follows from the Krasovskii-LaSalle invariance theorem that all trajectories of the system (78)–(79) converge to the largest invariant set M contained within the set Tn × Γ, where Γ ⊂ Rn is from (46). We next show that again M = Φ × Γ, just as in the proof of Theorem 2. It follows from (79) that this systems also satisfies q T γ˙ ≡ 0 and thus admits a foliation whose leaves are the invariant manifolds Tn × Ξr . Because B T Σ(γ) is the constant zero on the set Γ, its derivative is zero on M: 0 ≡ B T Σ0(γ)γ˙ ≡ B T Σ0(γ)QQT γ˙
− 1 ≡ B TRS T Σ0(γ)CS S T C 2 S 2 QT γ˙ .

(81)

Because the columns of B TR and S are independent and because the diagonal matrix Σ0(γ)C is positive definite, it follows that QT γ˙ ≡ 0 and thus γ˙ ≡ 0 on M. It then follows from (79) that e(ϕ) ≡ 0 on M, and we conclude that M ⊂ Φ × Γ. We observe that Φ × Γ is itself invariant under the dynamics (78)–(79), and it follows that M = Φ × Γ. We now fix r ∈ R and examine the dynamics on the invariant manifold Tn × Ξr . Using (62), we see that the derivatives of the projected state variables (63) along trajectories of (78)–(79) are w˙ 1 = S T Σ(Qw2 + αr )

(82)

w˙ 2 = QTCe(Rw1 ) − QTCBH(Qw2 + αr )B T Σ(Qw2 + αr ) .

(83)

As in the proof of Theorem 2, the equilibria of this system are precisely all points of the form (µ? , 0) for vectors µ? ∈ S T Φ. The equilibrium with µ? = 0 represents a fully synchronized trajectory with δϕ ≡ 0, and all other equilibria represent out-of-phase, frequency-synchronized trajectories. Because M = Φ×Γ, because Γ ∩ Ξr is a singleton, and because the points of S T Φ are isolated, we see that each trajectory of the system (82)–(83) converges to an equilibrium. We now perform the local linearization analysis at each equilibrium (µ? , 0) of the system (82)–(83). Similarly, to (73)–(74) we can approximate (82)–(83) around (µ? , 0) using w˙ 1 ≈ Zw2

(84)

w˙ 2 ≈ −YL[ (µ? )(w1 − µ? ) − YL[η Zw2

(85)

21

where L[η (αr ) = RT Lη (αr )R

Lη (αr ) = BH(αr )B T ,

and

(86)

with the matrices Lη (αr ) and L[η (αr ) also satisfying a condition analogous to (52). We can now use a dual version of Theorem 6 based on [36, Theorem 5, Corollary 2] which we state below. Lemma 8. Suppose Λ ∈ Cq×q has no eigenvalues on the imaginary axis, suppose H ∈ Cq×q is an invertible Hermitian matrix, and suppose ΛH + HΛ? > 0, where Λ? denotes the conjugate transpose of Λ. Then the number of eigenvalues of Λ having positive real part is the same as the number of positive eigenvalues of H. The dual version of Theorem 6 is then: Theorem 9. Let Λ ∈ R2p×2p be the block matrix 

0

Λ= −YL1

Z −YL2 Z

 ,

(87)

where L1 , L2 , Y, Z ∈ Rp×p satisfy: 1) L1 is symmetric and invertible, 2) L2 is symmetric with L2 > 0, 3) Y is symmetric and invertible, and 4) Y −1Z is symmetric with Y −1Z > 0. If L1 has a strictly negative eigenvalue, then Λ has an eigenvalue with a strictly positive real part. If instead L1 > 0, then Λ is Hurwitz. The proof is in Appendix C. Thus, we can use Theorem 7 together with Theorem 9 to show that, for the system (84)–(85), the equilibrium with µ? = 0 is exponentially stable and any other equilibria with µ? 6= 0 is exponentially unstable. The rest of the proof follows that of Theorem 2. VI. Concluding remarks We have presented two distributed controllers for the phase and frequency synchronization of heterogeneous nonlinear oscillators, and we have shown that each guarantees almost-global convergence on arbitrary connected graphs. Our solutions can be readily implemented using analog oscillators and phase comparators, and our analysis holds under very general assumptions on the system. In particular, unlike most existing work, we neither require the set of admissible frequencies to be unbounded, nor assume any special network topology.

22

Appendix A. Proof of Theorem 4 We let T denote the finite collection of all m × m diagonal matrices ∆ = diag{δ1 , . . . , δm } such that for all k ∈ E, either δk = f 0(0) or δk = f 0(π). For each such matrix ∆ ∈ T, we define the closed set 
P∆ = a ∈ Rm : det RTB diag(a)∆B TR = 0 ,

(88)

where diag(a) = diag{a1 , . . . , am } denotes the diagonal matrix whose diagonal entries are the m elements of a. Now ∆ is invertible by assumption (A5), and furthermore the columns of B TR are independent; it follows that P∆ 6= Rm (take diag(a) = ∆−1 ), which means P∆ is a closed algebraic set having zero measure. Thus P=

[

P∆

(89)

∆∈T

is also a closed algebraic set having zero measure. Therefore the set O = Rm \ P is a nonempty open semialgebraic set. Next we define the mapping H : Tn−1 × O → Rn−1 by H(µ, a) = RTB diag(a)F (B TRµ) .

(90)

The Jacobian matrix of H is  DH(µ, a) =

∂H (µ, a) ∂µ

 ∂H (µ, a) , ∂a

(91)

where ∂H (µ, a) = RTB diag(a)F 0(B TRµ)B TR ∂µ

(92)


∂H (µ, a) = RTB diag F (B TRµ) . ∂a

(93)

If we define the matrix 

I




   J(µ, a) =  diag(a) diag F (B TRµ) +  ,     · F 0(0) − F 0(B TRµ) B TR

(94)

where (·)+ denotes the Moore–Penrose pseudoinverse, then DH(µ, a) · J(µ, a) = RTB diag(a)∆(µ)B TR ,

(95)

where ∆(µ) is the diagonal matrix

+ ∆(µ) = F 0(B TRµ) + diag F (B TRµ) diag F (B TRµ)   · F 0(0) − F 0(B TRµ) .

(96)

It follows from assumptions (A4)–(A5) that f (θ) = 0 if and only if θ ∈ {0, π}, so for any µ ∈ Tn−1 , the matrix ∆(µ) in (96) belongs to T. It follows from the definition of O that the matrix in (95) is invertible, and

23

we conclude that DH(µ, a) has rank n − 1 for all (µ, a) ∈ Tn−1 × O. Thus H t {0},2 and it follows from the parametric transversality theorem3 [37, Theorem 6.35] that there exists a set Y ⊂ O having zero measure such that if a ∈ O \ Y then Ha t {0}, where Ha denotes the mapping µ 7→ H(µ, a). Choose Z = P ∪ Y; we have thus shown that for all a ∈ Rm \ Z, if µ is such that H(µ, a) = 0, then the matrix in (92) is invertible. Suppose a ∈ Rm is the edge weight vector so that A = diag(a), suppose a ∈ Rm \ Z, and suppose µ ∈ S T Φ. Then Rµ ∈ Φ which means H(µ, a) = 0, and it follows that L[ (µ) in (50), which is the same as the matrix in (92), is invertible. B. Proof of Theorem 6 Because both Y and Y −1Z are symmetric matrices, we see that the following holds: Y −1ZY = Y −1Z


T

Y = Z T Y −1 Y = Z T .

(97)

We will investigate the stability of the zero solution of the system x˙ = Λx. Partition the state as x = [xT1 xT2 ]T with x1 , x2 ∈ Rp , and consider the quadratic function Υ(x) = Υ(x1 , x2 ) = xT1 Lx1 + xT2 Y −1Zx2

(98)

whose derivative along trajectories of x˙ = Λx is ˙ Υ(x) = −2xT1 LXLx1 + 2xT1 LZx2 − 2xT2 Y −1ZY Lx1 = −xT1 L(X + X T )Lx1 6 0 .

(99)

First suppose L has an eigenvalue λ < 0 with unit eigenvector v. The standard Chetaev instability conditions [38, Theorem 4.3] do not guarantee exponential instability, so we will instead exploit the linear-quadratic structure to strengthen the Chetaev result. The cone  C = x ∈ R2p : Υ(x) < 0 ,

(100)

Υ(v, 0) = v T Lv = λv T v = λ < 0 .

(101)

0 6 xT2 Y −1Zx2 < −xT1 Lx1 = xT1 Lx1 ,

(102)

|Υ(x)| 6 xT1 Lx1 + xT2 Y −1Zx2 < 2 xT1 Lx1 .

(103)

is nonempty because

For any x ∈ C we have

which means

2If M and N are smooth manifolds, if f : N → M is smooth, and if S is an embedded submanifold of M , then f is transverse to S, written f t S, when for every p ∈ f −1 (S) we have Tf (p) M = Tf (p) S + dfp (Tp N ), where dfp denotes the differential of f at p. 3This transversality theorem is based on Sard’s theorem, which holds only for sufficiently smooth functions. One can verify that in our case, this application of Sard’s theorem is valid when the phase coupling function f is merely C 1 .

24

Let L+ denote the Moore–Penrose pseudoinverse of L, which is nonzero because L has a nonzero eigenvalue λ. Then we have T x1 Lx1 = xT1 LL+Lx1 6 kL+ k · kLx1 k2 .

(104)

Let σ > 0 denote the minimum eigenvalue of X + X T , so that 2 ˙ Υ(x) = −xT1 L(X + X T )Lx1 6 −σkLx1 k .

(105)

It follows that for any x ∈ C we have ˙ Υ(x)
0; then the largest invariant set contained in the set {Υ(x) = 0} is the origin x = 0. In this case Υ is positive definite, hence asymptotic stability follows from the KrasovskiiLaSalle invariance theorem. C. Proof of Theorem 9 For the instability part of Theorem 6, we do not require the matrices L1 or Z to be invertible, even though they happen to be so in our application. However, the above Chetaev-style proof of Theorem 6 does not carry over to a proof of the instability part of Theorem 9; instead, we will apply Lemma 8. To apply this lemma to matrix Λ in (87), we must show that Λ has no eigenvalues on the imaginary axis. Suppose Λv = jλv for some λ ∈ R and v ∈ C2p , and partition v as v = [v1T v2T ]T with v1 , v2 ∈ Cp . Then Zv2 = jλv1 −YL1 v1 − YL2 Zv2 = jλv2 .

(107) (108)

If λ = 0, then v1 = v2 = 0 because L1 , Y , and Z are all invertible. Otherwise we solve for v2 in (107) and substitute into (108) to obtain   Y λ2 Y −1Z −1 − L1 − jλL2 v1 = 0 .

(109)

If we write v1 = x + jy for x, y ∈ Rp , then because Y is invertible we obtain   0 = λ2 Y −1Z −1 − L1 − jλL2 (x + jy)     = Xx + λL2 y + j −λL2 x + Xy ,

(110)

where we have defined X = λ2 Y −1Z −1 − L1 . Setting the imaginary part to zero in (110) and solving for x yields x = λ1 L−1 2 Xy .

(111)

REFERENCES

25

Setting the real part to zero in (110) and substituting (111) for x yields   0 = λ1 XL−1 2 X + λL2 y .

(112)

Now (??) implies ZY = YZ T , which means T  −1  (Y −1Z −1 )T = (ZY )−1 = (ZY )T  −1  −1 = YZ T = ZY = Y −1Z −1 ,

(113)

2 and we conclude that X is symmetric. Thus the matrix XL−1 2 X + λ L2 is symmetric and positive definite,

and therefore (112) implies y = 0. Hence x = 0 from (111), which means v1 = 0 and thus also v2 = 0 from (107). Therefore, if Λv = jλv for some λ ∈ R and v ∈ C2p then v = 0, and we conclude that Λ has no eigenvalues on the imaginary axis. Let H ∈ R2p×2p be the block diagonal matrix  −L−1 1 H= 0

0 −Z −1 Y

 ,

which is symmetric and invertible by assumption. Then a straightforward calculation yields   0 0  > 0, ΛH + HΛT =  0 2YL2 Y

(114)

(115)

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